Note on the fractional Mittag-Leffler functions by applying the modified Riemann-Liouville derivatives

Adem Kilicman,Wedad Saleh

doi:10.5269/bspm.44103

Abstract

In this article, the fractional derivatives in the sense of the modified Riemann-Liouville derivative is employed for constructing some results related to Mittag-Leffler functions and established a number of important relationships between the Mittag-Leffler functions and Wright function.

Highlights

Then we obtain the following relationAssume that α > 0,β > 0 and λ ∈ R, the following formula holds DαnEα (λxα) = λnEα (λxα) where n = 1, 2, 3, · · ·
This function plays a crucial role in classical calculus for α = 1, for α = 1 it becomes the exponential function, that is ex = E1(x) ex =
This function plays an important role in the solution of a linear partial differential equation

Summary

Then we obtain the following relation

Assume that α > 0,β > 0 and λ ∈ R, the following formula holds DαnEα (λxα) = λnEα (λxα) where n = 1, 2, 3, · · ·. Assume that α > 0,β > 0 and λ ∈ R, the following formula holds DβnEα (λxα) = λnx(α−β)nEα,αn−βn+1(λxα). Assume that α > 0,β > 0 and λ ∈ R, the following formula holds. Assume that ηi > 0 are arbitrary real numbers and 0 < γ < 1, the following formula holds.

Similarly we can show that

Conclusion